Further monotonicity properties of renewal processes

Kijima, Masaaki

doi:10.2307/1427480

Cited by 13 publications

(6 citation statements)

References 13 publications

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“…Other monotonicities in the continuous-time setting seem not to be considered very often. See for example Kijima (1992) for the successful use of these monotonicities in the discrete-time case.…”

Section: Some Preliminariesmentioning

confidence: 99%

Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains

Kijima

1998

Journal of Applied Probability

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A continuous-time Markov chain on the non-negative integers is called skip-free to the right (left) if only unit increments to the right (left) are permitted. If a Markov chain is skip-free both to the right and to the left, it is called a birth–death process. Karlin and McGregor (1959) showed that if a continuous-time Markov chain is monotone in the sense of likelihood ratio ordering then it must be an (extended) birth–death process. This paper proves that if an irreducible Markov chain in continuous time is monotone in the sense of hazard rate (reversed hazard rate) ordering then it must be skip-free to the right (left). A birth–death process is then characterized as a continuous-time Markov chain that is monotone in the sense of both hazard rate and reversed hazard rate orderings. As an application, the first-passage-time distributions of such Markov chains are also studied.

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Section: Some Preliminariesmentioning

confidence: 99%

Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains

Kijima

1998

Journal of Applied Probability

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“…In fact, Baxter (1988) has shown that (27) holds for some processes that need not be ordinary renewal processes. Some related results can be found in Kijima (1992…”

Section: Moshe Shaked and Haolong Zhumentioning

confidence: 96%

“…Thus, the relationships among the three statements in the right-hand sides of (6), (7) and (5) are of interest. In Theorem 1 it is shown that these three statements are equivalent.…”

Section: Replacement Interval Tmentioning

confidence: 99%

Some results on block replacement policies and renewal theory

Shaked

Zhu

1992

Journal of Applied Probability

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Age and block replacement policies are commonly used in order to reduce the number of in-service failures. The focus in this paper is on the block replacement policies, about which relatively less is known than age replacement policies. Several new results which connect the properties of block replacement policies with the properties of the corresponding renewal function and the excess lifetimes are obtained. Some applications and the relationships between these new results and some known results are included.

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“…Brown's conjecture in the continuous case, however, has remained open. See Szekli (1986Szekli ( ), (1990, Hansen and Frenk (1991), Shaked and Zhu (1992), Kijima (1992), and Kebir (1997) for related results and discussions. Also relevant is the work of Lund et al (2006), who used hazard rates and renewal sequences to study reversible Markov chains.…”

Section: Introductionmentioning

confidence: 99%